A vessel in a plant where I work is the frustum of a cone on its side.
A liquid is contained in this section and pours out the end of the
cone section, therefore the liquid only takes up a certain portion of
the cone's volume. How can I compute the volume of the liquid?

Consider a frustum with a lower circular base radius equal to R, and
an upper circular base radius and the height both equal to R/2. If the
center of the lower base is at the origin, evaluate the triple
integral...

Each piano has a unique tuning curve that can be approximated through
3-point quadratic interpolation. Is there a formula that will allow me
to measure three different notes on the piano and find the tuning
curve in between these notes?

What is the formula for the volume of the shape that remains when a
cone is sliced by a plane parallel to the height of the cone, and the
chord created by the slice on the base circle is less than the diameter?

In the classic related rates problem where a ladder slides down a
wall, why are the rates of the top and bottom of the ladder different?
It seems to me that if you pull the bottom at a constant rate, the
top should also slide down the wall at the same constant rate.

I just learned integration by substitution. Part of my textbook's
explanation seems to depend on treating differentials as numbers that
cancel out. But I've also seen that differentials cannot be treated
that way. Can you clarify that and explain why substitution works?